290 X EXISTENCE THEOREMS
of 0 and such that jc(0) = 0 (use contradiction, and consider, in a compact interval
containing 0, the points where a solution reaches its maximum or minimum).
2. Let /(/, x) be the real valued continuous function defined in R2 by the following con-
ditions: /(/, x) = -2/ for x ^ t2, /(f, x) = -2xj t for \x\ < t2J(t, x) - 2t for x ^ ~t2.
Let (yn) be the sequence of functions defined by y0(t) = t2, yn(t) = J f(u> yn.i(u))du
for n ^ 1 . Show that the sequence (/„(/)) is not convergent for any / ^ 0, although the
differential equation x' =/(/, x) has a unique solution such that x(Q) = 0 (Problem 1).
3. For any pair of real numbers a > 0, /2 > 0, the function equal to — (t — a)2 for t < a,
to 0 for a ^ f ^ /?, to (/ - jS)2 for / > /3, is a solution of the differential equation *' =
2|*|1/2 such that X0) = 0.
Let u0 be an arbitrary continuous function defined in a compact interval [# , b], and
fr
define by induction «„(/) = 2 |wn-.i(s)|'/2fifr for fe[0,6]. Show that if y is the
J a
largest number in [a, b] such that uQ(t) = 0 in [0, y], the sequence (un) converges uni-
formly in [a, b] to the solution of x' = 2\x\1'2 which is equal to 0 for a ^ / ^ y, to
(/ — y)2 for y ^ / ^ b. (Consider first the case in which «0(0 = 0 for / < y, uQ(t) —
k(( _ y)2 for y^t^b. Next remark that, replacing if necessary UQ by «i, one may
suppose that UQ is increasing in [a, 6]; observe that for any number e > 0, there are
two numbers ki > 0, k2 > 0 such that in [a, b]
- y - e) ^ w0 (0 ^ £2 t?o(' — y + e)
wherei?0(/) = 0 if r^ 0, u0(0 == /2 if r ^ 0.)
4. The notations being those of Section 10.4, suppose K = R, /is continuous and bounded
in I x H, and let M = sup \\f(t, x)\\. Let *0 be a point of H, S an open ball of
<t,x)elxH
center XQ and radius r, contained in H.
(a) Suppose in addition/is uniformly continuous in I x S (a condition which is auto-
matically satisfied if E is finite dimensional and I is contained in a compact interval
I0 such that/is continuous in I0 x H). Prove that for any e > 0, and any compact
interval [/0, *o + h] (resp.[/0 — h, t0]) contained in I and such that h < r/(M + fi),
there exists in that interval an approximate solution of x/ ^f(t, x) with approximation
e, taking the value x0 for t = tQ. (Suppose S > 0 is such that the relations |fi — t2\ ^ S,
ll*i — *2IK 8 imply ||/(fi, *j) — f(t2, #2) II ^ e; consider a subdivision of the interval
Ifo9 fo + h] in intervals of length at most equal to inf(S, 8/M), and define the approxi-
mate solution on each successive subinterval, starting from /0.)
(b) Suppose E is finite dimensional and I = ]t0 — a, fQ -j- a[. Prove that there exists a
solution of x' =/(/, x), defined in the interval [/0, 'o -f c] (resp.[/0 — c, /0]) with c =
inf(a, r/M), taking its values in S, and equal to *0 for t = t0. (*' Peano's theorem": for
each 72, let un be an approximate solution with approximation l/n, defined in Jn»
l/o, /o H- c — (l//i)L whose existence is given by (a). Observe that for each m, the restric-
tions of the functions un (for n ^ m) to Jm form a relatively compact subset of the normed
normed space ^E(Jm) (7.5.7), and use the " diagonal process " as in the proof of (9.13.2);
finally apply (10.4.3) and (8.7.8).)
5. Let/be the mapping of the space (c0) of Banach (Section 5.3, Problem 5) into itself, such
that, for x = (*„),/(*) = (yn)t with yn = |xn|1/2 -f-------. Show that / is continuous in
n H- 1
(c0), but that there is no solution of the differential equation x' =/(*), defined in a
neighborhood of 0 in R, taking its values in (c0), and equal to 0 for t = 0. (If there was